I am reading differential topology and reading about the local immersion and local submersion theorem.
In the proof they made an assumption of the following:
let $h: R^n \rightarrow R^m$ be a injective linear map with $n \leq m$, then by a change of coordinate on $R^m$, we may assume $h$ has matrix representation of the form {$\frac{I_n}{0}$}. Which I take it as after we apply a change of coordinate on $R^n$ (choosing a different basis) we can assume the matrix representation of $h$ has that form.
Similarly if $f: R^n \rightarrow R^m$($n \geq m$) is subjective, linear then by C.O.C on $R^n$ we can assume $f$ has a matrix representation {${I_n | 0}$}.
I don't understand why we must apply the change of coordinate on the domain(codomain) for injective (surjective) linear map instead of the other way around? Is it arbitrary?